A A Highly Subjective Summary of the Gravitational Lens Time Delays to Date and the Derived
Values of the Hubble Constant

All discussion is based on parametric models (though see the text for non-parametric models). For
B0218+357 the mass-model is reasonably well known, thanks to an Einstein ring and VLBI structure [176]
but the galaxy centre astrometry is not. York et al. [178] find 70 5 (2) based on an ACS
determination of the galaxy centre, and Wucknitz et al. [176] find 78 6 (2) using the
ring and VLBI structure. I double the errors and take 74 5. For HE0435–1223 the image
positions of [174] with the lens redshift of [99] and the longest time delay of [79], together with an
isothermal sphere model with external shear, gives H0 = 90, similar to the value inferred by
Kochanek et al. A 20% error for the lack of knowledge of the mass profile and another 20%
for the presence of nearby galaxies are added in quadrature to the time delay error to give
H0 = 90 26. For SBS0909+532 the system is very underconstrained, especially as the flux ratios
are not well known, possibly due to extinction [166]. Lehár et al. [88] find a predicted time
delay of 49 38 days for H0 = 70, by considering the allowed space where 1, the
time delay of 45 days [166] translates into H0 = 76 59; one might argue that the errors
should be slightly smaller but this lens is not in any case going to contribute significantly to
the overall total. In RXJ0911+0551 the situation is complicated by the presence of a nearby
cluster. Hjorth et al. [55] obtain 71 9 (2) based on a velocity dispersion for the cluster
from [74] (but see also discussion in [77]). As with B0218+357 I double the errors and take
71 9. In FBQ0951+2635 an isothermal model gives [68]. The same source claims that
in this lens the variation of H0 with mass slope is relatively weak, but I nevertheless add an
extra 20% to the error and take 60 15. For Q0957+561, a system with a huge cluster
contribution, probably the most compelling analysis (of the vast literature on this object) is that
in [70] where the lensed host galaxy is used to disentangle the cluster shear from the internal
shear produced by the lensed galaxy shape. Assuming a cluster convergence of 0.2, using the
range of Keeton et al.’s sample models and increasing the errors by 20% gives 90 25. For
SDSS J1004+4112 the cluster contribution makes modelling so problematic that it is difficult to
sort out sensible H0 estimates from simple models. However, once the cluster mass profile is
thoroughly understood using the large number of constraints and the remaining time delays are
known, this may give the best H0 estimate of all [42]. HE1104–185 is also a bizarre system in
which the closer image is brighter; Lehár et al. [88] predict much larger time delays than are
actually found, and typical isothermal models plus the time delay in [101] give H0 95. For
consistency I adopt this value, together with a 20% error added in quadrature to the time delay
error to reflect the fact that we do not know the galaxy mass profile, giving 95 25. In
PG1115+080 traditional fits of isothermal models give H0 in the forties [135], but a more recent
stellar dynamical measurement [159] suggests a steeper-than-isothermal profile and with the
time delay in [135] gives H0 = , so I adopt 59 12. In RXJ1131–1231 almost
any smooth isothermal model fails (and H0 10), so Morgan et al. [98] suggest that a
large piece of substructure or satellite galaxy falls close to one of the images. While plausible,
this means that the system is almost impossible to model for H0. There is a measured time
delay for JVAS B1422+231 [110] but in my view it needs confirmation before being used for H0
determination. SBS 1520+530 is modelled in [18], who also provide the time delay, by an isothermal
model together with consideration of a nearby cluster. I add 20% to their error, due to the
isothermal assumption, and get 51 13. In B1600+434 an isothermal model gives
(random errors [81], but adjusted for a ( = 0.3, 0.7) model). I add 20% for systematics
involved in lack of knowledge of the mass profile and get 60 20. In B1608+656 Koopmans
et al. [83] assemble an impressive array of data, including three time delays, stellar velocity
measurements and Einstein ring fitting to get , to which I add another 10% due to a feeling
of unease about the lens being two interacting galaxies and consequent effects on the mass
profile, and take 75 10. For SDSS J1650+4251 I adopt the value of H0 = 52 from the
time delay and isothermal models of [167], plus an extra 20% error for the mass slope, and use
H0 = 52 11. In PKS1830–211 models in [173] using their improved galaxy centre position
give 44 9, to which I add the now traditional 20% for ignorance of the mass slope and
10% (cf. B1608+656) for the possibility of associated secondaries [25], to give 44 13.
Finally, for HE2149-2745, a simple SIE model gives H0 = 49, in agreement with the average of
quoted in [76] on analysis of this plus four other lenses; Burud et al. [17] derive a higher
value due to restrictions on assumed ellipticity of the lens galaxy. Adding the usual 20% for
ignorance of the mass slope to the time delay error we obtain 49 11. At last, the overall
average is 66 3 (or 61 4 without B0218+357). We should also consider the effect
of mass along the line of sight, which is likely to add a further 5 – 10% to the error budget
of each lens system and which will probably not reduce as ; much of it is systematic if
the error is normally in the sense of ignoring nearby groups of matter which add to the local
convergence.

It should be emphasised that this is an attempt to work out one observer’s view of the situation based
on parametric models, and that the reader will have noticed a lot of subjective judgements and arbitrary
manipulation of errors going on, but you did ask.